Making Progress during a Stall in the Simplex Algorithm.

All the standard methods for avoiding cycling in the simplex algorithm set up a subprogram related to the blocking rows in the canonical form and look for an improving homogeneous solution. The method proposed in this paper does not. Instead, the Gass-Saaty parametric method applied to the objective cost form is used to choose the incoming column. The pivot-row choice among blocking rows, i.e., those tied for pivot, is arbitrary. A simple anti-cycling device is used which avoids dual degeneracy of the parameterized objective with probability one. Tests were run on nine highly degenerate practical test problems ranging in size from small to large. Using MINOS software, the standard simplex method required 31,195 iterations of which 20,504 or 66 were blocked pivots. Using the parametric scheme required 14,379 iterations, of which 9,481 or 66 were also blocked. The reduction in the number of iterations for this set of highly degenerate test problems is over 50. CPU time reduction under the current way of implementation is 40.